Related papers: Non-commutative width and Gopakumar-Vafa invariant…
Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that…
We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of…
We initiate explicit computations of Vafa-Witten invariants of 3-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa-Witten invariants of 3-manifolds in a family of concrete…
Associated with a prime homology class $\beta \in P_2(X,\Z)$ (i.e. $\beta=p\alpha$ and $\alpha \in H_2(X,\Z)$ imply $p=1$ or $p$ is an odd prime) on a symplectic three-manifold with vanishing first Chern class, we count the embedded…
A difference equation is proved for the Gromov-Witten potential of the resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten invariants of any Calabi-Yau threefold, it is further shown that similar difference…
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the…
We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces…
We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The…
We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces.…
This is an expository article on the noncommutative singularity theory of power series in noncommuting variables, its motivation from deformation theory, and its applications to contractibility of curves and the classification of smooth…
This paper aims to continue the classification of non-smooth regular curves, but over fields of characteristic three. These curves were originally introduced by Zariski as generic fibers of counterexamples to Bertini's theorem on the…
We compute Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge2$. The genus zero invariants have been computed a long ago, Gromov-Witten invariants of any genus for del Pezzo surfaces of degree $\ge3$ have been found…
We prove that genus zero Gromov--Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus zero orbifold Gromov--Witten invariants of an appropriate root stack construction along the divisor.
Quadratic Gromov--Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field $k$ without assuming that $k$ is the field of complex or real numbers. This paper studies the behavior…
Based on computations of Pandharipande, Zinger proved that the Gopakumar-Vafa BPS invariants $\mathrm{BPS}_{A,g}(X,\omega)$ for primitive Calabi-Yau classes and arbitrary Fano classes $A$ on a symplectic $6$-manifold $(X,\omega)$ agree with…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants.…
For even dimensional smooth complete intersections, of dimension at least 4, of two quadric hypersurfaces in a projective space, we study the genus zero Gromov-Witten invariants by the monodromy group of its whole family. We compute the…
We construct characteristic-zero lifts of partial Hasse invariants for genus zero non-compact curves in Hilbert modular varieties. The construction is based on recent results on the associated Picard-Fuchs differential equations. As an…
Motivated by the real version of the Gopakumar-Vafa conjecture for 3-folds, the authors introduced in [GI] the notion of local real Gromov-Witten invariants associated to local 3-folds over Real curves. This article is devoted to the proof…